A number of the form $x + iy$ where x and y are real numbers and i$^{2}$ = -1 or i = $\sqrt{-1}$ is called a complex number. $x$ is called the real part of the complex number and y is called its imaginary part. The set of all complex numbers is denoted by $C$ = {$x + iy$ |x, y $\epsilon $ $R$}. |

Any point on the x-axis has coordinates of the form $(x, 0)$ and hence represents the complex number $x + i0$ which is a purely real number. Hence the x - axis is referred to as the real axis of the complex plane. Any point on the y axis has coordinates of the form $(0, y)$ and represents a complex number of the form $0 + iy$ which is a purely imaginary number.

Therefore, the y- axis is referred to as the imaginary axis. The origin has coordinates (0, 0) and hence it represents the complex number 0 + i 0.

The diagram in which the complex numbers are represented by points of the complex plane is called Argand's diagram. The point

$(x, y)$ represents $Z$ = $x + iy$ and its image in the real axis is $(x, -y)$ and hence represents the complex number $x - iy$ = $\bar{Z}$.

If $Z$ = $x + iy$ is a complex number, then $x - iy$ is called the conjugate of $Z$ and is denoted by $\bar{Z}$, and Z and $\bar{Z}$ are called conjugate complex numbers.

**The following results follow if $Z$ and $\bar{Z}$ are conjugate complex numbers:**

**1)**$Z$ + $\bar{Z}$ = $(x + iy)$ + $(x - iy)$ = 2$x$

**2)**$x$ = Real part of $Z$ is $\frac{Z + \bar{Z}}{2}$

**3)**$y$ = Imaginary part of $Z$ is $\frac{Z - \bar{Z}}{2i}$

**4)**$Z$$\bar{Z}$ = $(x + iy)$ - $(x - iy)$ = x$^{2}$ + y$^{2}$ = |z|$^{2}$

**Operations on complex numbers are explained below:**

Addition of two complex numbers:

If $Z_{1}$ = $x_{1} + iy_{1}$ and $Z_{2}$ = $x_{2} + iy_{2}$ are two complex numbers, then their sum denoted by $Z_{1} + Z_{2}$ is defined to be the complex number ($x_{1} + x_{2}$) + i (y$_{1} + y_{2}$)

Multiplication of complex numbers:

If $Z_{1}$ = $x_{1} + iy_{1}$ and $Z_{2}$ = $x_{2} + iy_{2}$ then their product denoted by $Z_{1}Z_{2}$ is defined to the complex number $Z_{1}Z_{2}$ = ($x_{1}x_{2} - y_{1}y_{2}) + i(x_{1}y_{2}+x_{2}y_{1}$)

Division of a complex number by a non zero complex number:

If $Z_{1}$ = $x_{1} + iy_{1}$ and $Z_{2}$ = $x_{2} + iy_{2}$ $\neq$ 0 then

$\frac{Z_{1}}{Z_{2}}$ = $\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}$

= $\frac{x_{1}+iy_{1}}{x_{2}+iy_{2}}$ $\times$ $\frac{x_{2}-iy_{2}}{x_{2}-iy_{2}}$

= $\frac{(x_{1}x_{2}+y_{1}y_{2})+i(y_{1}x_{2}-y_{2}x_{1})}{x_{2}^{2}+y_{2}^{2}}$

which is a complex number

Modulus of a complex number:

If $Z$ = $x + iy$ is a complex number, then the modulus of $Z$ denoted by $|Z|$ is the real number $\sqrt{x^{2}+y^{2}}$

$|Z|$ = $|x + iy|$ = $\sqrt{x^{2}+y^{2}}$

$|Z|$ = $|x - iy|$ = $\sqrt{x^{2}+y^{2}}$ = $| x - iy|$ = $| -Z|$

$|Z|$ = $|-Z|$

**Example 1:**Express $\frac{1}{4 + 3i}$ in the form $x + iy$

**Solution :**

$\frac{1}{4 + 3i}$ = $\frac{1}{4+3i}$ $\times$ $\frac{4-3i}{4-3i}$

= $\frac{4 - 3i}{16 + 9}$

= $\frac{4}{25}$ + ($\frac{-3}{25}$)$i$

= $\frac{4}{25}$ - $i$ ($\frac{3}{25}$)

**Example 2:**Find the modulus of $\frac{1}{(2+i)^{2}}$ - $\frac{1}{(2-i)^{2}}$

**Solution:**

$\frac{1}{(2+i)^{2}}$ - $\frac{1}{(2-i)^{2}}$ = $\frac{1}{3+4i}$ - $\frac{1}{3-4i}$

= $\frac{3-4i}{(3+4i)(3-4i)}$ - $\frac{3+4i}{(3-4i)(3+4i)}$

= $\frac{3-4i}{9+16}$ - $\frac{3+4i}{9+16}$

= $\frac{3-4i-3-4i}{25}$

=$\frac{-8i}{25}$

= 0 - $\frac{8}{25}$

Its modulus = $\sqrt{0+\frac{64}{625}}$ = $\frac{8}{25}$